The Yinggehai Basin is an important Cenozoic gas bearing basin in the South China Sea. With the gradual improvement of gas exploration and over-development in shallow layers, deep overpressured layers have become the main target for natural gas exploration. There are no large-scale faults in the strata above the Meishan Formation in the central depression, and hydraulic fracturing caused by overpressure in mudstone cap rocks is the key factor for the vertical differential distribution of gas. In this paper, based on the leak-off data, pore fluid pressure, and rock mechanics parameters, the Fault Analysis Seal Technology (FAST) method is used to analyze the hydraulic fracture risk of the main mudstones in the central depression. The results show that the blocks in the diapir zone have been subjected to hydraulic fracturing in the Huangliu cap rocks during the whole geological history, and the blocks in the slope zone which is a little distant from the diapirs has a lower overall risk of hydraulic fracture than the diapir zone. In geological history, the cap rocks in slope zone remained closed for a longer time than in diapir zone and being characterized by the hydraulic fracture risk decreases with the distance from the diapirs. These evaluation results are consistent with enrichment of natural gas, which accumulated in both the Yinggehai Formation and Huangliu Formation of the diapir zone, but it only accumulated in the the Huangliu Formations of the slope zone. The most reasonable explanation for the difference of the gas reservoir distribution is that the diapirs promote the development of hydraulic fractures: (1) diapirism transfers deep overpressure to shallow layers; (2) the small fault and fractures induced by diapir activities weakened the cap rock and reduced the critical condition for the natural hydraulic fractures. These effects make the diapir zone more prone to hydraulic fracturing, which are the fundamental reasons for the difference in gas enrichment between the diapir zone and the slope zone.